Dynamical self-similarity, $L^{q}$-dimensions and Furstenberg slicing in $\mathbb{R}^d$
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916384964870144 |
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| author | Corso, Emilio Shmerkin, Pablo |
| author_facet | Corso, Emilio Shmerkin, Pablo |
| contents | We extend a theorem of the second author on the $L^q$-dimensions of dynamically driven self-similar measures from the real line to arbitrary dimension. Our approach provides a novel, simpler proof even in the one-dimensional case. As consequences, we show that, under mild separation conditions, the $L^q$-dimensions of homogeneous self-similar measures in $\mathbb{R}^d$ take the expected values, and we derive higher rank slicing theorems in the spirit of Furstenberg's slicing conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_04608 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dynamical self-similarity, $L^{q}$-dimensions and Furstenberg slicing in $\mathbb{R}^d$ Corso, Emilio Shmerkin, Pablo Classical Analysis and ODEs Combinatorics Dynamical Systems Metric Geometry We extend a theorem of the second author on the $L^q$-dimensions of dynamically driven self-similar measures from the real line to arbitrary dimension. Our approach provides a novel, simpler proof even in the one-dimensional case. As consequences, we show that, under mild separation conditions, the $L^q$-dimensions of homogeneous self-similar measures in $\mathbb{R}^d$ take the expected values, and we derive higher rank slicing theorems in the spirit of Furstenberg's slicing conjecture. |
| title | Dynamical self-similarity, $L^{q}$-dimensions and Furstenberg slicing in $\mathbb{R}^d$ |
| topic | Classical Analysis and ODEs Combinatorics Dynamical Systems Metric Geometry |
| url | https://arxiv.org/abs/2409.04608 |